between 1 and 10 the logs. will be between 0 and 1
„ 10 „ 100 „ „ 1 „ 2
„ 100 „ 1000 „ „ 2 „ 3
„ 1000 „ 10,000 „ „ 3 „ 4
In other words, the logarithms of numbers between 1 and 10 will be
wholly fractional (_i.e._, decimal); the logs. of numbers between 10 and
100 will be 1 _followed by a decimal quantity_; the logs. of numbers
between 100 and 1000 will be 2 followed by a decimal quantity, and so
on. These decimal quantities for numbers from 1 to 10 (which are the
logarithms of this particular series) are as follows:—
Numbers 1 2 3 4 5 6 7 8 9 10
Logarithms 0 0·301 0·477 0·602 0·699 0·778 0·845 0·903 0·954 1·000
Combining the two tables, we can complete the logarithms. Thus for 3
multiplied successively by 10, we have:—
Numbers 3 30 300 3000 30,000 etc.
Logarithms 0·477 1·477 2·477 3·477 4·477 „
We see from this that for numbers having the _same significant figure_
(or figures), 3 in this case, the decimal part or _mantissa_ of the
logarithm is the same, but that the integral part or _characteristic_ is
always _one less than the number of figures before the decimal point_.
For numbers less than 1 the same plan is followed. Thus extending our
first table downwards, we have:—
Numbers 1 0·1 0·01 0·001 0·0001 etc.
Logarithms 0 −1 −2 −3 −4 „
so that for 3 divided successively by 10, we have:—
Numbers 3 0·3 0·03 0·003 0·0003 etc.
Logarithms 0·477 ̅1·477 ̅2·477 ̅3·477 ̅4·477 „
Here again we see that with the same significant figures in the numbers,
the mantissa of the logarithm has always the same (_positive_) value,
but the characteristic is _one more_ than the _number of 0’s immediately
following the decimal point_, and is _negative_, as indicated by the
minus sign written over it. Only the decimal parts of the logarithms of
numbers between 1 and 10 are given in the usual tables, for, as shown
above, the logarithms of all tenfold multiples or submultiples of a
number can be obtained at once by modifying the characteristic in
accordance with the rules given.
An examination of the two rows of figures giving the logarithms of
numbers from 1 to 10 will reveal some striking peculiarities, and at the
same time serve to illustrate the principle of logarithmic calculation.
First, it will be noticed that the addition of any two of the logarithms
gives the logarithm of the _product_ of these two numbers. Thus, the
addition of log. 2 and log. 4 = 0·301 + 0·602 = 0·903, and this is seen
to be the logarithm of 8, that is, of 2 × 4. Conversely, the difference
of the logarithms of two numbers gives the logarithm of the _quotient_
resulting from the division of these two numbers. Thus, log. 8 − log. 2
= 0·903 − 0·301 = 0·602, which is the log. of 4, or of 8 ÷ 2.
One other important point is to be noted. If the logarithm of any number
is _multiplied_ by 2, 3, or any other quantity, whole or fractional, the
result is the logarithm of the original number, raised to the 2nd, 3rd,
or other power respectively. Thus, multiplying the log. of 3 by 2, we
obtain 0·477 × 2 = 0·954, and this is seen to be the log. of 9, that is,
of 3 raised to the 2nd power, or 3 _squared_. Again, log. 2 multiplied
by 3 = 0·903—that is, the log. of 8, or of 2 raised to the 3rd power, or
2 _cubed_. Conversely, dividing the logarithm of any original number by
any number _n_, we obtain the logarithm of the _n_th root of the
original number. Thus, log. 8 ÷ 3 = 0·903 ÷ 3 = 0·301, and is therefore
equal to log. 2 or to the log. of the _cube root_ of 8.
Only simple logs. have been taken in these examples, but the student
will understand that the same reasoning applies, whatever the number.
Thus for 20^3 we prefix the characteristic (1 in this case) to log. 2,
giving 1·301. Multiplying by 3, we have 3·903 as the resulting
logarithm, and as its characteristic is 3, we know that it corresponds
to the number 8000. Hence 20^3 = 8000.
In this brief explanation is included all that need now be said with
regard to the properties of logarithms. The main facts to be borne
clearly in mind are:—(1.) That to find the _product_ of two numbers, the
logarithms of the numbers are to be _added_ together, the result being
the logarithm of the product required, the value of which can then be
determined. (2.) That in finding the _quotient_ resulting from the
division of one number by another, _the difference_ of the logarithms of
the numbers gives the logarithm of the quotient, from which the value of
the latter can be ascertained. (3.) That to find the result of _raising
a number to the nth power_, we _multiply_ the logarithm of the number by
_n_, thus obtaining the logarithm, and hence the value, of the desired
result. And (4.) That to find the n_th root of a number_, we _divide_
the logarithm of the number by _n_, this giving the logarithm of the
result, from which its value may be determined.